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User blog:Edwin Shade 2/Extension Of Luxius' Exploding Array Notation (EOLEAN)
See here **''Anything that follows is based on Luckyluxius/Luckyluxiuz's Exploding array notation. I credit him for the basis of this notation and the general ideas contained herewithin.'' Explanation Luckyluxius's Exploding Array Notation (LEAN) becomes unclear after the first seven examples given, and therefore I'll be generalizing only on the beginning part. Namely, the following: :\(\{2,0,(2,3)\} = 2\) :\(\{2,1,(2,3)\} = 2 + 3 + 2 + 3\) :\(\{2,2,(2,3)\} = 2 \cdot 3 \cdot 2 \cdot 3\) :\(\{2,3,(2,3)\} = 2^{3^{2^{3}}}\) :\(\{2,4,(2,3)\} = 2 \uparrow 3 \uparrow 2 \uparrow 3\) :\(\{2,5,(2,3)\} = 2 \uparrow\uparrow 3 \uparrow\uparrow 2 \uparrow\uparrow 3\) :\(\{2,n,(2,3)\} = 2 \uparrow^{n-3} 3 \uparrow^{n-3} 2 \uparrow^{n-3} 3\) (if n > 3) From this, I figured that the first number in the array (in all these cases, 2), refered to how many times the elements in the string (2,3) repeated, seperated by \(\uparrow^{n-3}\). Consequently, I thought up an extension based on this, such that instead of two elements being in the third entry, there could be an arbitrary number of entries, like this: :\(\{a,b,(e_0,e_1,e_2,...,e_n)\}\) Defining EOLEAN First, let \(\Delta_n\) refer to the nth hyper-operator, such that \(a\Delta_0b = a+1\), \(a\Delta_1b = a+b\), \(a\Delta_2b = a\cdot b\), \(a\Delta_3b = a^b\), etc. In general, \(a\Delta_{n+1}b = \underbrace{a\Delta_n a\Delta_n a\Delta_n...\Delta_n a\Delta_n a\Delta_n a}_{\text{b occurrences of a}}\), where \(a\Delta_0b = a+1\). These hyper-operators are right-binding of course, meaning that \(a\Delta_{n+1}b = \underbrace{a\Delta_n a\Delta_n a\Delta_n...\Delta_n a\Delta_n a\Delta_n a}_{\text{b occurrences of a}}\) is implicitly the same as \(a\Delta_{n+1}b = \underbrace{a\Delta_n (a\Delta_n (a\Delta_n...\Delta_n (a\Delta_n (a\Delta_n a))))}_{\text{b occurrences of a}}\). With this definition, we can phrase "\(\{2,n,(2,3)\} = 2 \uparrow^{n-3} 3 \uparrow^{n-3} 2 \uparrow^{n-3} 3\) (if n > 3)" as the following: :\(\{2,n,(2,3)\} = 2\Delta_{n-1}3\Delta_{n-1}2\Delta_{n-1}3\) "(2,3)" can be any two numbers, e0 and e1, in which case a more general equation would be: :\(\{2,n,(e_0,e_1)\} = e_0\Delta_{n-1}e_1\Delta_{n-1}e_0\Delta_{n-1}e_1\) Now to generalize the initial 2 in {2,n,(e0,e1)} to any number, we must define a secondary function, \(g_x\). We will set \(g_0 = e_0\Delta_{n-1}e_1\) and consequently say that \(g_x =e_{x\;(mod\;2)}\Delta_{n-1}g_{x-1}\). This generates a hyperpower tower where e0 and e1 alternate. Since there are two numbers which alternate once per cycle, this means that \(g_{2\cdot(x-1)}\) is equal to \(\{x,n,(e_0,e_1)\}\). There is only one more generalization to go, and that is on the number of elements contained by parenthesis in the third entry. We need to define \(\{x,n,(e_0,e_1,e_2,...,e_y)\}\). To do this, we will define a set Ey such that the nth element in Ey (denoted \(\in_n(E_y)\)) is equal to \(e_{y-n+1\;(mod\;y+1)}\). This is because I do not want to have to bother with modulo functions, so it's easier to define an infinite set out of which we can pick elements that are ordered according to our objectives than have to work with modulos, which I don't really like. Now, we may define our new secondary function as follows: :\(h_z = \in_z(E_y)\) Then a tertiary function as follows: :\(j_v = h_v\Delta_{n-1} j_{v-1}\text{ where }j_0 = h_0\) Finally, the complete general definition: :\(\{x,n,(e_0,e_1,e_2,...,e_y)\} = j_{xy}\) ---- The above is need of correction, review is pending. ---- Category:Blog posts